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Teacher Guide: Histograms

Learn how to read and create histograms to display continuous data distributions.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Statistical Graphs. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Explain the purpose of a histogram and how it differs from a bar graph
  • Read and interpret frequency data from a histogram
  • Calculate the total number of data points from histogram frequencies
  • Create appropriate intervals for a given data set
  • Describe the shape of a distribution (symmetric, skewed)
Prerequisites
  • • Understanding of numerical data vs categorical data
  • • Ability to read bar graphs
  • • Basic understanding of frequency and counting
  • • Familiarity with number ranges and intervals
Discussion Starters
  • 1. Why do you think histogram bars touch while bar graph bars have gaps?
  • 2. If a histogram is shaped like a mountain with the peak in the middle, what does that tell us about the data?
  • 3. How would a histogram of ages at a retirement home look different from ages at a school?
  • 4. What questions can a histogram help us answer that a simple list of numbers cannot?
Common Misconceptions

Thinking taller bars mean higher values

Remediation: Clarify that bar height shows frequency (how many), not the data values themselves. A tall bar at 20-30 means many data points in that range, not that the values are large.

Believing all graphs with bars are the same

Remediation: Compare side-by-side: bar graph (categories, gaps) vs histogram (intervals, no gaps). Emphasize the difference in data types.

Differentiation Ideas

For Struggling Students:

  • • Provide pre-made interval templates for students to fill in
  • • Use physical manipulatives to sort data into bins before graphing
  • • Start with small data sets (10-15 points) with clear patterns

For On-Level Students:

  • • Create histograms from raw data with 20-30 data points
  • • Compare two histograms and describe differences
  • • Calculate percentages of data in specific intervals

For Advanced Students:

  • • Explore how changing interval width affects the histogram shape
  • • Analyze real-world data sets from the internet
  • • Investigate bimodal distributions and what causes them
Standards Alignment
  • 6.SP.B.4 (CCSS.MATH.CONTENT.6.SP.B.4)

    Display numerical data in plots on a number line, including dot plots, histograms, and box plots

  • 6.SP.B.5 (CCSS.MATH.CONTENT.6.SP.B.5)

    Summarize numerical data sets in relation to their context

Lesson Resources
  • visualInteractive Histogram Builder

    Students create histograms by dragging data points into bins

  • activityClass Height Histogram

    Measure and plot student heights to create a real histogram

  • worksheetHistogram vs Bar Graph Sort

    Classify graphs and identify which data type each represents

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A histogram is a type of graph that shows the distribution of numerical data. It uses bars to display how often data values fall within specific ranges called intervals or bins.
Key features of a histogram:
  • The x-axis shows the intervals (ranges of values)
  • The y-axis shows the frequency (how many data points fall in each interval)
  • Bars touch each other (no gaps!) because the data is continuous
Frequency=Number of data points in an interval\text{Frequency} = \text{Number of data points in an interval}

Worked Examples

A histogram shows test scores. The interval 70−8070-80 has a bar reaching up to 1212. What does this tell us?
1

Identify the interval

The interval is 70−8070-80, meaning scores from 70 up to (but not including) 80 → Score range: 70-79
2

Read the frequency

The bar height is 1212 on the y-axis → Frequency = 12
3

Interpret the meaning

12 students scored between 70 and 79 points → 12 students in this range

Common Mistakes

Leaving gaps between bars

Why it's wrong: In histograms, bars touch because the data is continuous. Gaps would imply missing data ranges.

Correct: Always draw histogram bars touching each other. Bar graphs (for categorical data) have gaps; histograms do not.

Confusing histograms with bar graphs

Why it's wrong: Bar graphs show categories (like favorite colors). Histograms show continuous numerical ranges.

Correct: Histogram = continuous data with intervals, bars touch. Bar graph = categories, bars have gaps.

Using unequal interval widths

Why it's wrong: Unequal widths make it hard to compare frequencies fairly - a wider bar naturally captures more data.

Correct: Keep all intervals the same width (e.g., all 10 units wide: 0-10, 10-20, 20-30).

Miscounting which interval a value belongs to

Why it's wrong: Boundary values can be confusing. Is 20 in the 10-20 or 20-30 interval?

Correct: Use a consistent rule: intervals include the lower bound but exclude the upper bound. So 20 goes in 20-30, not 10-20.

Why It Matters

Histograms help us understand how data is distributed:
  • Test Scores: Teachers use histograms to see how a class performed - are most scores clustered in the middle, or spread out?
  • Heights: Scientists study height distributions to understand growth patterns in populations
  • Wait Times: Businesses analyze customer wait times to improve service
  • Weather: Meteorologists display temperature ranges to show climate patterns
Unlike simple lists of numbers, histograms give us an instant visual picture of our data!

Real World Applications

Analyzing Test Scores

Teachers use histograms to understand class performance at a glance.

Example:

A histogram of 30 test scores shows: 0-50 (2 students), 50-60 (5), 60-70 (8), 70-80 (10), 80-90 (4), 90-100 (1).

1Try It Yourself

Looking at the histogram above, which interval has the most students?

What is the frequency of the most common interval?

Step 1: Write the mathematical expression

Find the highest bar in the histogram:

Customer Wait Times

Restaurants track how long customers wait to be seated to improve service.

Example:

A histogram shows wait times: 0-5 min (15 customers), 5-10 min (25), 10-15 min (18), 15-20 min (7), 20+ min (3).

2Try It Yourself

Using the wait time histogram, find the total number of customers surveyed.

How many customers were included in this data?

Step 1: Write the mathematical expression

Add all the frequencies: 15+25+18+7+315 + 25 + 18 + 7 + 3

Student Heights in PE Class

Physical education teachers track student heights to group students fairly for activities.

Example:

Heights in cm: 140-150 (4), 150-160 (12), 160-170 (18), 170-180 (8), 180-190 (3).

3Try It Yourself

What percentage of students are between 160 and 170 cm tall?

Calculate the percentage of students in the 160-170 cm range.

Step 1: Write the mathematical expression

Use the formula: (frequency÷total)×100(\text{frequency} \div \text{total}) \times 100

Key Takeaways

  • 1A histogram displays the distribution of continuous numerical data using bars
  • 2The x-axis shows intervals (bins), the y-axis shows frequency (count)
  • 3Bars in a histogram touch each other because the data is continuous
  • 4The total number of data points equals the sum of all frequencies
  • 5The shape of a histogram reveals patterns: symmetric, left-skewed, or right-skewed

Frequently Asked Questions

What is the difference between a histogram and a bar graph?

A histogram shows continuous numerical data with touching bars, while a bar graph shows categorical data with gaps between bars. Histograms use intervals; bar graphs use distinct categories.

How do I choose the number of intervals?

A common guideline is 5-15 intervals. Too few intervals hide details; too many create noise. The square root of the number of data points is a good starting point.

What if a data value falls exactly on a boundary?

Use a consistent rule. Typically, intervals include the lower bound but exclude the upper bound. So 50 would go in the 50-60 interval, not the 40-50 interval.

Glossary

Histogram
A graph that shows the distribution of numerical data using bars where each bar represents an interval
Frequency
The number of data points that fall within a specific interval
Interval (Bin)
A range of values used to group data in a histogram (e.g., 10-20, 20-30)
Distribution
The pattern of how data values are spread across different intervals
Skewed
When data is not symmetric and leans more toward one side

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